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Tooth Bending Strength of Spur and Helical Gears

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Gears

Part of the book series: Springer Series in Solid and Structural Mechanics ((SSSSM,volume 11))

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Abstract

In this chapter, a general survey is first done on the tooth bending strength of spur and helical gears, also focusing attention on fatigue tooth root breakage and safety factor to be used for their design. The theoretical bases of calculation of tooth bending strength are then discussed, with particular reference to the constant strength parabola introduced by Lewis and 30° tangent lines proposed and used subsequently. Stress state at the tooth root is then analyzed, considering the load application at the outer point of single pair gear tooth contact as well as the local stress concentrations , also due to combined local notch effects. Finally, the procedure for calculating the tooth bending strength of these types of gears in accordance with the ISO standards is described, highlighting when deemed necessary as the relationships used by the same ISO are founded on the theoretical bases previously discussed.

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References

  • Abdullah MQ, Jweeg MJ (2012) Analytical solution of bending stress equation for symmetric and asymmetric involute gear teeth shapes with and without profile correction. Innov Syst Des Eng 3(6):19–33

    Google Scholar 

  • Almen JO (1935) Factors influencing the durability of spiral bevel gears for automobiles. Automot Ind 73:662–668 and pp 696–701

    Google Scholar 

  • Almen JO, Straub JC (1937) Factors influencing the durability of automobile transmission gears, Automot Ind 77, 25th Sept, pp 426–432, and 9th Oct, 1937, pp 488–493

    Google Scholar 

  • Aziz IAA, Idris DMN, Ghazali WM (2017) Investigation bending strength of spur gear: a review. In: MATEC web of conferences, 90:01037, AiGEV 2016

    Google Scholar 

  • Becker AA (1992) The boundary element method in engineering: a complete course. McGraw-Hill Book Company Inc, New York

    Google Scholar 

  • Brebbia CA (1978) The boundary element method for engineers. Pentech Press, New York

    Google Scholar 

  • Brebbia CA, Connor JJ (1973) Fundamentals of finite element techniques. Butterworths, London

    Google Scholar 

  • Brooks RE (2016) An introduction to shot peening for increasing gear fatigue life. Gear Solut Mag

    Google Scholar 

  • Buckingham E (1949) Analytical mechanics of gears. McGraw-Hill Book Company Inc, New York

    Google Scholar 

  • Burr AH (1982) Mechanical analysis and design. Elsevier Science Publishing Co., Inc, New York

    Google Scholar 

  • Castigliano CAP (1935) SELECTA, a cura di Gustavo Colonnetti. R. Luigi Avalle Editore, Torino

    Google Scholar 

  • Castigliano CAP (1966) The theory of equilibrium of elastic systems and its applications, translated from Théorie de l’Équilibre des Systèmes Élastiques et ses Applications by Ewart S. Dover Publications Inc, Andrews, New York

    Google Scholar 

  • Castigliano CAP (1984) SELECTA 1984, a cura di Edoardo Benvenuto e Vittorio Nascé. Editrice Levrotto&Bella, Torino

    Google Scholar 

  • Chauhan V (2016) A review on effect of some important parameters on the bending strength and surface durability of gears. Int J Sci Res Publ 6(3):289–298

    Google Scholar 

  • Collins JA (1993) Failure of materials in mechanical design: analysis, prediction, prevention, 2nd edn. John Wiley & Sons Inc, New York

    Google Scholar 

  • Cook RD (1981) Concepts and applications of finite element analysis, 2nd edn. John Wiley & Sons Inc, New York

    MATH  Google Scholar 

  • Dieter GE (1988) Mechanical metallurgy, adapted by David Bacon, London: McGraw-Hill Book Company (UK) Limited

    Google Scholar 

  • Dolan TJ, Broghammer EL (1942) A photoelastic study of stresses in gear tooth fillets, University of Illinois Bulletin, XXXIX (31), Engineering Experiment Station Bulletin Series No. 335

    Google Scholar 

  • Faupel JH (1964) Engineering design: a synthesis of stress analysis and material engineering. John Wiley & Sons Inc, New York

    Google Scholar 

  • Fenner RT (1986) Engineering elasticity: application of numerical and analytical techniques. Ellis Horwood Limited Publishers, Chichester

    Google Scholar 

  • Fernandes PJL (1996) Tooth bending fatigue failures in gears. Eng Fail Anal 3(3):219–225

    Article  Google Scholar 

  • Frost NE, Marsh KJ, Pook LP (1974) Metal fatigue. Oxford University Press, London

    Google Scholar 

  • Fuchs HO, Stephens RI (1980) Metal fatigue in engineering. John Wiley & Sons Inc, New York

    Google Scholar 

  • Galilei G (1638) Discorsi e Dimostrazioni Matematiche intorno a due nuove scienze attinenti alla Meccanica & i Movimenti Locali. In: Leida, Appresso gli Elsevirii, M.D.C. XXXVIII

    Google Scholar 

  • Garro A, Vullo V (1978a) Alcune considerazioni sul proporzionamento degli ingranaggi, Atti del VI Convegno Nazionale AIAS, Brescia, 22–24 giugno

    Google Scholar 

  • Garro A, Vullo V (1978b) Note integrative sulla memoria: Alcune considerazioni sul proporzionamento degli ingranaggi, Atti del VI Convegno Nazionale AIAS, Brescia, 22–24 giugno

    Google Scholar 

  • Garro A, Vullo V (1979) Acoustic problems in vehicle transmissions, Nauka I Motorna Vozila ‘79, Science and Motor Vehicles ‘79, Jugoslavija, 4–7 jun

    Google Scholar 

  • Gere JM, Timoshenko SP (1997) Mechanics of materials, 4th edn. PWS Publishing Company, Boston

    Google Scholar 

  • Giovannozzi R (1965a) Costruzione di macchine, vol I, 2nd ed. Casa Editrice Prof. Riccardo Pàtron, Bologna

    Google Scholar 

  • Giovannozzi R (1965b) Costruzione di macchine, vol. II, 4th ed. Casa Editrice Prof. Riccardo Pàtron, Bologna

    Google Scholar 

  • Henriot G (1970) French gear rating practices. Paper presented in commemoration of the presentation of the Edward P. Connell Award to Emeritus Professor Dr. Ing. Gustav Niemann at the AGMA Meeting, Oct., St. Louis, Mo

    Google Scholar 

  • Henriot G (1979) Traité théorique et practique des engrenages 1, 6th edn. Bordas, Paris

    Google Scholar 

  • Heywood RB (1952) Designing by photoelasticity. Chapman & Hall Ltd, London

    Google Scholar 

  • Heywood RB (1962) Designing against fatigue. Chapman & Hall Ltd, London

    Google Scholar 

  • Ishida K (1977) Computer simulation of stresses and deformations of gear case, and of gear teeth taking the influence of gear body into consideration. World Congress on Gearing, Paris 22–24 June, pp 309–323

    Google Scholar 

  • ISO 54:1996 Cylindrical gears for general engineering and foe heavy engineering—Modules

    Google Scholar 

  • ISO 6336-3:2006 Calculation of load capacity of spur and helical gears—part 3: calculation of tooth bending strength

    Google Scholar 

  • ISO/TR 6336-30:2017 Calculation of load capacity of spur and helical gears—part 30: calculation examples for the application of ISO 6336 parts 1, 2, 3, 5

    Google Scholar 

  • Juvinall RC (1967) Engineering considerations of stress, strain, and strength. McGraw-Hill Book Company, New York

    Google Scholar 

  • Juvinall RC, Marshek KM (2012) Fundamentals of machine component design, 5th edn. John Wiley & Sons Inc, New York

    Google Scholar 

  • Kapelevich AL (2013) Direct gear design. CRC Press, Taylor & Francis Group, Boca Raton, Florida

    Book  Google Scholar 

  • Kapelevich AL, Shekhtman YV (2003) Direct gear design: bending stress minimization. Gear Technol pp 44–47

    Google Scholar 

  • Kelley BW, Pedersen R (1956) The beam strength of modern gear-tooth design. Vortragauf der Fachtagung Antriebselemente, Essen

    Google Scholar 

  • Kelley BW, Pedersen R (1958) The beam strength of modern gear-tooth design. SAE Tech Paper 580017

    Google Scholar 

  • Kramberger J, Šraml M, Glodez S, Flašker J, Potrč I (2004) Computational model for the analysis of bending fatigue in gears. Comput Struct 82(23):2261–2269

    Article  Google Scholar 

  • Kuhn P (1964) The prediction of notch and crack strength under static or fatigue loading. SAE-ASME Paper 843C

    Google Scholar 

  • Lewis W (1892) Investigation of Strength of Gear Teeth. In: Proceedings of Engineers Club, Philadelphia, USA, October pp 16–23, and vol. 10 January 1893

    Google Scholar 

  • Li S (2007) Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications. Mech Mach Theory 42(1):88–114

    Article  MATH  Google Scholar 

  • Li S (2008) Effect of addendum on contact strength, bending strength and basic performance parameters of a pair of spur gears. Mech Mach Theory 43(12):1557–1584

    Article  MATH  Google Scholar 

  • Lipson C, Juvinall RC (1963) Handbook of stress and strength. The Macmillan Company, New York

    Google Scholar 

  • McMullen FE, Durkan TM (1922) The gleason works system of bevel gears. Machinery

    Google Scholar 

  • Merritt HE (1954) Gears, 3 ed., Sir Isaac Pitman & Sons, Ltd, London

    Google Scholar 

  • Miyachika K, Oda S (1991) Bending strength of internal spur gears. In: Proceedings of the international conference on motion and power transmissions, Hiroshima, Japan, November 23–26, pp 781–786

    Google Scholar 

  • Neuber H (1946) Theory of Notch Stresses. In: Edwards JW (eds) Publisher, Inc., Ann Arbor, Mich., (translation of the original German version in 1937)

    Google Scholar 

  • Niemann G (1965) Maschinenelemente Entwerfen, Berechnen und Gestalten im Maschinenbau, vol 2. Springer-Verlag, Getriebe, Berlin Heidelberg

    Google Scholar 

  • Niemann G, Richter W (1954) Tragfähigste evolventen-schragverzahnung. Vieweg, Zahnräder, Zahnradgetriebe, Braunschweig

    Google Scholar 

  • Niemann G, Richter W (1960) Versuchsergebnisse zur Zahnflankentragfähigkeit, 9 Publications in Z. Konstruktion, S. 185, 236, 269, 319, 360

    Google Scholar 

  • Niemann G, Winter H (1983) Maschinen-Elemente, Band II: Getriebe allgemein, Zahnradgetriebe-Grundlagen, Stirnradgetriebe. Springer Verlag, Berlin Heidelberg

    Google Scholar 

  • Niemann G, Winter H, Höhn BR (2005) Maschinenelemente - Band 1: Konstruktion und Berechnung von Verbindungen, Lagern, Wellen, 4. Auflage. Springer-Verlag, Berlin Heidelberg

    Google Scholar 

  • Paul FW, Faucett TR (1962) The superposition of stress concentration factors. J Manuf Sci Eng 84(1):129–134

    Google Scholar 

  • Pedrero JI, Vallejo II, Pieguezuelos M (2007) Calculation of tooth bending strength and surface durability of high transverse contact ratio spur and helical gear drives. ASME J Mech Des 129(1):69–74

    Article  Google Scholar 

  • Peterson RE (1953) Stress concentration design factors. John Wiley & Sons Inc, New York

    Google Scholar 

  • Peterson RE (1962) Fatigue of metals in engineering and design. Edgard Marburg Lecture, ASTM, Philadelphia

    Google Scholar 

  • Peterson RE (1974) Stress concentration factors. John Wiley & Sons Inc, New York

    Google Scholar 

  • Püchner O, Kamenski A (1972) Spannungskonzentration und Kerbwirkung von Kerben im Kerbrand. Konstruktion 24(4):127–134

    Google Scholar 

  • Savage M, Rubadeux KL, Coe HH (1995) Bending strength model for internal spur gear teeth, NASA technical memorandum 107012. In: 31st joint propulsion conference and exhibit, San Diego, California, July 10–12

    Google Scholar 

  • Shipley EE (1967) Gear failures: how to recognize them, what causes them, how to avoid them. Mach Des

    Google Scholar 

  • Straub JC (1953) Shot peening in the design of gears. In: Proceedings of the annual meeting of the american gear manufacturers association. AGMA, Hot Spring, VA (Virginia), USA, May 31-June 3

    Google Scholar 

  • Timoshenko SP (1953) History of strength of materials. McGraw-Hill Book Company Inc, New York

    Google Scholar 

  • Timoshenko SP (1955) Strength of materials. D. Van Nostrand Co, Part I, Princeton, New Jersey

    MATH  Google Scholar 

  • Timoshenko SP (1956) Strength of materials. D. Van Nostrand Co, Part II, Princeton, New Jersey

    MATH  Google Scholar 

  • Timoshenko SP, Goodier JN (1951) Theory of elasticity, 2nd edn. McGraw-Hill Book Company Inc, New York

    MATH  Google Scholar 

  • Ugural AC (2015) Mechanical design of machine components, 2nd edn. CRC Press, Taylor & Frencis Group, Boca Raton, Florida

    Google Scholar 

  • Vijayakar SM, Houser DR (1988) The use of boundary elements for the determination of the AGMA geometry factor. Gear Technology, January/February

    Google Scholar 

  • Vullo V (2014) Circular cylinder and pressure vessels, stress analysis and design. Springer International Publishing Switzerland, Cham Heidelberg

    Google Scholar 

  • Vullo V, Maisano G (1981) Metodi normalizzati per il calcolo delle ruote dentate. Progettare 8–9:57–64

    Google Scholar 

  • Vullo V, Vivio F (2013) Rotors: stress analysis and design. Springer-Verlag Italia, Milan Dordrecht

    Book  MATH  Google Scholar 

  • Wellauer EJ (1970) Comments on German, French and AGMA gear rating practices. Paper presented in commemoration of the presentation of the Edward P. Connell Award to Emeritus Professor Dr. Ing. Gustav Niemann at the AGMA Meeting, Oct., St. Louis, Mo

    Google Scholar 

  • Wellauer EJ, Seireg A (1960) Bending strength of gear teeth by cantilever-plate theory. ASME J Eng Ind 82(3):213–220

    Article  Google Scholar 

  • Wilcox L, Coleman W (1973) Application of finite elements to the analysis of gear tooth stresses. ASME, J Eng Ind 95(4):1139–1148

    Article  Google Scholar 

  • Winter H (1961) Gear tooth strength of spur gears. Power Transm 404, 460, 516

    Google Scholar 

  • Winter H (1962) Gear tooth strength of spur gears, Power Transm 66, 124

    Google Scholar 

  • Winter H, Hösel Th (1969) Tragfähigkeitsberechnung von Stirnund Kegelrädern nach DIN 3990. VDI-Z 111:209

    Google Scholar 

  • Winter H, Stolzle K (1970) German gear rating practices. Paper presented in commemoration of the presentation of the Edward P. Connell Award to Emeritus Professor Dr. Ing. Gustav Niemann at the 1970 AGMA Meeting, Oct., St. Louis, Mo

    Google Scholar 

  • Zienkiewicz OC (1977) The finite element method, 3rd edn. McGraw-Hill Higher Education, UK

    MATH  Google Scholar 

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Vullo, V. (2020). Tooth Bending Strength of Spur and Helical Gears. In: Gears. Springer Series in Solid and Structural Mechanics, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-030-38632-0_3

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  • DOI: https://doi.org/10.1007/978-3-030-38632-0_3

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